M Theory Lesson 22

The little disc diagrams of the last post are elements of the 2-discs 1-operad. To each $n \in \mathbb{N}$ we associate the diagram with $n$ distinct little discs inside the bigger disc. There is a $d$-discs 1-operad in any dimension $d$.

It is natural to wonder whether or not the boundary nature of operad mathematics can tell us anything about holography (this is Cambridge?). This has been discussed, such as in this article by Zois, which is motivated by Kontsevich's original idea that the physical principle should be related to the higher dimensional Deligne conjecture. This conjecture now has many proofs (apparently), most notably via the operad methods of Batanin.

In other words, what we really care about are $n$-operad analogues of the 2-discs operad. Recall that 2-categorical TFTs were studied by Pfeiffer and Lauda. With objects, 1-arrows and 2-arrows, there is room for both open and closed strings. But rather than their stringy diagrams, we would like to invent higher operads of discs with extra data.

Moving up in dimension to 3-spheres (for building quaternions) we would naturally consider 3-sphere diagrams with embedded knots and links. The whole diagram is a 2-arrow, whereas the 3-sphere pieces are 1-arrows between the links as their local boundary. The nice thing about knots is that they have primes.

It sounds like we're doing homotopy theory of spheres, doesn't it? Well, that is in fact one way of looking at higher category theory, only we need to attach arrows to our diagrams.